Two automated assembly robots, Robot P and Robot Q, produce identical components. Robot P … — GRE General Test

Two automated assembly robots, Robot P and Robot Q, produce identical components. Robot P operates at a constant rate of $p$ components per hour, and Robot Q operates at a constant rate of $q$ components per hour, where $p > q > 0$ . During a shift, Robot P worked for $4$ hours and Robot Q worked for $6$ hours to produce a combined total of $360$ components. If $T$ represents the total number of components produced when Robot P works for $7$ hours and Robot Q works for $3$ hours, which of the following statements must be true? Select all such statements.

The rate of Robot P, $p$ , must be greater than $36$ components per hour.Cevap
The rate of Robot Q, $q$ , must be less than $36$ components per hour.Cevap
The total number of components $T$ must be greater than $360$ and less than $630$ .Cevap
D
The rate of Robot P, $p$ , must be less than $60$ components per hour.
E
The total number of components $T$ could be equal to $320$ .

Cevap

The correct statements are: the rate of Robot P,

p

, must be greater than

36

components per hour; the rate of Robot Q,

q

, must be less than

36

components per hour; and the total number of components

T

must be greater than

360

and less than

630

The system of equations

2p + 3q = 180

combined with

p > q > 0

strictly bounds

p

between

36

and

90

, and

q

between

0

and

36

. Substituting these boundary conditions into the total expression

T = 5p + 180

yields the strict range

360 < T < 630

. Consequently, the three statements asserting

p > 36

q < 36

, and

360 < T < 630

are all mathematically required.

Adım Adım Çözüm

Set up the linear equation from the initial production shift and simplify.

4p + 6q = 360 \implies 2p + 3q = 180 \implies q = 60 - \frac{2}{3}p

This establishes the exact relationship between the production rates

p

and

q

Apply the given constraints

p > q > 0

to determine the domain bounds for

p

and

q

p > 60 - \frac{2}{3}p \implies \frac{5}{3}p > 60 \implies p > 36

. Also,

q > 0 \implies 60 - \frac{2}{3}p > 0 \implies p < 90

. Thus,

36 < p < 90

and

0 < q < 36

Determining extreme bounds for

p

automatically constrains both individual rates.

Formulate

T

in terms of

p

and evaluate its numerical boundaries.

T = 7p + 3q = 7p + (180 - 2p) = 5p + 180

. Substituting

36 < p < 90

gives

360 < T < 630

Substituting

3q = 180 - 2p

simplifies

T

into a single-variable linear modeling equation.

Anahtar Kavram

Linear word problem modeling, variable elimination, and system inequality constraint analysis

Tahmini Süre:1m 45s

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